Near-resonances and detuning in classical and quantum mechanics

G. Gaeta; G. Pucacco; G. Gaeta; G. Pucacco

doi:10.3934/mine.2023005

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-44. doi: 10.3934/mine.2023005

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Near-resonances and detuning in classical and quantum mechanics

G. Gaeta ^{1
,
,},
G. Pucacco ^{2
,
,}

1.
Dipartimento di Matematica, Università degli Studi di Milano, v. Saldini 50, 20133 Milano, Italy
2.
Dipartimento di Fisica and INFN, Sezione di Roma Ⅱ, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy

Received: 22 May 2021 Revised: 06 December 2021 Accepted: 07 December 2021 Published: 17 January 2022

From the point of view of perturbation theory, (perturbations of) near-resonant systems are plagued by small denominators. These do not affect (perturbations of) fully resonant systems; so it is in many ways convenient to approximate near resonant systems as fully resonant ones, which correspond to considering the "detuning" as a perturbation itself. This approach has proven very fruitful in Classical Mechanics, but it is also standard in (perturbations of) Quantum Mechanical systems. Actually, its origin may be traced back (at least) to the Rayleigh-Ritz method for computing eigenvalues and eigenvectors of perturbed matrix problems. We will discuss relations between these approaches, and consider some case study models in the different contexts.
- perturbation theory,
- detuning,
- classical mechanics,
- quantum mechanics,
- resonance,
- near-resonance
Citation: G. Gaeta, G. Pucacco. Near-resonances and detuning in classical and quantum mechanics[J]. Mathematics in Engineering, 2023, 5(1): 1-44. doi: 10.3934/mine.2023005

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Abstract

From the point of view of perturbation theory, (perturbations of) near-resonant systems are plagued by small denominators. These do not affect (perturbations of) fully resonant systems; so it is in many ways convenient to approximate near resonant systems as fully resonant ones, which correspond to considering the "detuning" as a perturbation itself. This approach has proven very fruitful in Classical Mechanics, but it is also standard in (perturbations of) Quantum Mechanical systems. Actually, its origin may be traced back (at least) to the Rayleigh-Ritz method for computing eigenvalues and eigenvectors of perturbed matrix problems. We will discuss relations between these approaches, and consider some case study models in the different contexts.

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